From exp (coefficients) to Odds Ratio and their interpretation in Logistic Regression with factors

Question

I ran a linear regression of acceptance into college against SAT scores and family / ethnic background. The data are fictional. This is a follow-up on a prior question, already answered. The question focuses in the gathering and interpretation of odds ratios when leaving the SAT scores aside for simplicity.

The variables are Accepted (0 or 1) and Background ("red" or "blue"). I set up the data so that people of "red" background were more likely to get in:

fit <- glm(Accepted~Background, data=dat, family="binomial")
exp(cbind(Odds_Ratio_RedvBlue=coef(fit), confint(fit)))

                        Odds_Ratio_RedvBlue             2.5 %       97.5 %
(Intercept)             0.7088608                     0.5553459   0.9017961
Backgroundred           2.4480042                     1.7397640   3.4595454

Questions:

1. Is 0.7 the odd ratio of a person of "blue" background being accepted? I'm asking this because I also get 0.7 for "Backgroundblue" if instead I run the following code:

   fit <- glm(Accepted~Background-1, data=dat, family="binomial")
   exp(cbind(OR=coef(fit), confint(fit)))
   

2. Shouldn't the odds ratio of "red" being accepted ($\rm Accepted/Red:Accepted/Blue$) just the reciprocal: ($\rm OddsBlue = 1 / OddsRed$)?

Answer 1

I've been working on answering my question by calculating manually the odds and odds ratios:

Acceptance   blue            red            Grand Total
0            158             102                260
1            112             177                289
Total        270             279                549

So the Odds Ratio of getting into the school of Red over Blue is:

$$ \frac{\rm Odds\ Accept\ If\ Red}{\rm Odds\ Acccept\ If\ Blue} = \frac{^{177}/_{102}}{^{112}/_{158}} = \frac {1.7353}{0.7089} = 2.448 $$

And this is the Backgroundredreturn of:

fit <- glm(Accepted~Background, data=dat, family="binomial")
exp(cbind(Odds_and_OR=coef(fit), confint(fit)))

                      Odds_and_OR                         2.5 %      97.5 %
(Intercept)             0.7088608                     0.5553459   0.9017961
Backgroundred           2.4480042                     1.7397640   3.4595454

At the same time, the (Intercept)corresponds to the numerator of the odds ratio, which is exactly the odds of getting in being of 'blue' family background: $112/158 = 0.7089$.

If instead, I run:

fit2 <- glm(Accepted~Background-1, data=dat, family="binomial")
exp(cbind(Odds=coef(fit2), confint(fit2)))

                        Odds            2.5 %      97.5 %
Backgroundblue     0.7088608        0.5553459   0.9017961
Backgroundred      1.7352941        1.3632702   2.2206569

The returns are precisely the odds of getting in being 'blue': Backgroundblue (0.7089) and the odds of being accepted being 'red': Backgroundred (1.7353). No Odds Ratio there. Therefore the two return values are not expected to be reciprocal.

Finally, How to read the results if there are 3 factors in the categorical regressor?

Same manual versus [R] calculation:

I created a different fictitious data set with the same premise, but this time there were three ethnic backgrounds: "red", "blue" and "orange", and ran the same sequence:

First, the contingency table:

Acceptance  blue    orange  red   Total
0             86        65  130     281
1             64        42  162     268
Total        150       107  292     549

And calculated the Odds of getting in for each ethnic group:

• Odds Accept If Red = 1.246154;
• Odds Accept If Blue = 0.744186;
• Odds Accept If Orange = 0.646154

As well as the different Odds Ratios:

• OR red v blue = 1.674519;
• OR red v orange = 1.928571;
• OR blue v red = 0.597186;
• OR blue v orange = 1.151717;
• OR orange v red = 0.518519; and
• OR orange v blue = 0.868269

And proceeded with the now routine logistic regression followed by exponentiation of coefficients:

fit <- glm(Accepted~Background, data=dat, family="binomial")
exp(cbind(ODDS=coef(fit), confint(fit)))

                      ODDS     2.5 %   97.5 %
(Intercept)      0.7441860 0.5367042 1.026588
Backgroundorange 0.8682692 0.5223358 1.437108
Backgroundred    1.6745192 1.1271430 2.497853

Yielding the odds of getting in for "blues" as the (Intercept), and the Odds Ratios of Orange versus Blue in Backgroundorange, and the OR of Red v Blue in Backgroundred .

On the other hand, the regression without intercept predictably returned just the three independent odds:

fit2 <- glm(Accepted~Background-1, data=dat, family="binomial")
exp(cbind(ODDS=coef(fit2), confint(fit2)))

                      ODDS     2.5 %    97.5 %
Backgroundblue   0.7441860 0.5367042 1.0265875
Backgroundorange 0.6461538 0.4354366 0.9484999
Backgroundred    1.2461538 0.9900426 1.5715814